Calculation method for electron collision ionization cross section of hot dense plasma

Abstract

The invention discloses a method for calculating an electron collision ionization cross section of a hot dense plasma, and belongs to the technical field of plasmas, and the method comprises the steps: S1, employing a super network chain to approximately calculate correlation functions of free electron-free electron, electron-ion and ion-ion coupling models; s2, introducing the correlation function into an FAC program, and calculating a bound electron wave function of the target element in a plasma environment; and S3, based on the bound electron wave function in the plasma environment, calculating the electron collision ionization cross section of the target element in the plasma environment by adopting twisted wave approximation. According to the technical scheme provided by the invention, the free electron shielding effect and the inter-ion coupling effect in the plasma environment are introduced into the FAC through the correlation function of the HNC approximate coupling model, so that the deviation caused by the collision of the free atom model on the ionization section in the description of the hot dense plasma is reduced, and the calculation precision is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of plasma, and particularly relates to a method for calculating the electron impact ionization cross-section of hot dense plasma. Background Art

[0002] In an ultra-high temperature and dense plasma environment, the coupling effect between the ion structure and the environment has an important impact on the basic parameters of the plasma, and the related effects cannot be ignored. In hot dense plasma, establishing a reasonable calculation method to describe the interaction between charged particles is the most important step in studying electron impact ionization.

[0003] The study of electron impact cross-sections in high-temperature and dense plasmas is an important topic in the fields of nuclear physics and plasma physics, and involves multiple application scenarios such as nuclear fusion, astrophysics, inertial confinement fusion, and high energy density physics. Plasma is an ionized gas composed of free electrons and ions. Under high-temperature and high-density conditions, the collision processes between electrons and ions, atoms, and other electrons have a decisive impact on the thermodynamic, radiative, and transport properties of the plasma. The electron impact cross-section is a key physical quantity that describes these collision processes and directly affects the energy exchange, ionization equilibrium, radiation spectrum, and transport coefficients of the plasma.

[0004] In nuclear fusion research, high-temperature and dense plasma is the core medium for achieving controllable nuclear fusion. For example, in magnetic confinement fusion devices (such as tokamaks) and inertial confinement fusion experiments, the plasma needs to be heated to a high temperature state of millions or even hundreds of millions of degrees Celsius. Under such extreme conditions, the collision frequency and energy transfer efficiency between electrons and ions directly affect the heating, confinement, and stability of the plasma. The accurate calculation of the electron impact cross-section is crucial for optimizing the fusion reaction conditions and improving the energy output efficiency.

[0005] In the field of astrophysics, high-temperature and dense plasmas widely exist in extreme astrophysical environments such as the interiors of stars, white dwarfs, neutron stars, and supernova explosions. The plasmas in these environments are usually in a state of local thermodynamic equilibrium or non-local thermodynamic equilibrium, and the electron collision processes dominate key physical processes such as radiative transport, energy balance, and element synthesis. For example, in the interior of a star, the electron impact ionization and excitation processes determine the formation of the radiation spectrum and the energy transport efficiency; in a supernova explosion, the electron impact cross-section directly affects the propagation of the shock wave and the synthesis of heavy elements.

[0006] In inertial confinement fusion and high energy density physics experiments, the hot and dense plasma generated by the interaction of laser or particle beams with target materials has extremely high energy density and complex dynamic behaviors. Electron collision processes play a central role in these experiments. For example, electron-ion collisions determine the heating rate and temperature distribution of the plasma, while electron-electron collisions affect the heat conduction and energy relaxation processes of the plasma. Accurate electron collision cross-section data is the basis for numerical simulations and experimental designs, and is of great significance for understanding plasma dynamics and optimizing experimental conditions.

[0007] However, calculating electron collision cross-sections in hot and dense plasmas faces many challenges. First, the strong coupling and quantum effects in the plasma make traditional perturbation theories (such as the Born approximation) no longer applicable, and more accurate theoretical models need to be developed, such as density functional theory, quantum Monte Carlo methods, and molecular dynamics simulations. Second, many-body effects, screening effects, and dynamic effects in hot and dense plasmas make the calculation of electron collision cross-sections extremely complex. In addition, directly measuring electron collision cross-sections in hot and dense plasmas experimentally also faces great difficulties, and indirect methods (such as spectroscopic diagnostics and X-ray scattering) are usually needed to verify the accuracy of theoretical models.

[0008] In the prior art, for the electron impact ionization process in a hot and dense plasma environment, the Debye-Hückel calculation method is usually used, which calculates the influence of the screening effect of free electrons in the plasma environment on the electron impact ionization of hydrogen-like ions. However, this method mainly focuses on the free electron screening effect and insufficiently considers the combined effect of the free electron screening effect and the ion-ion coupling effect in the hot and dense plasma environment, resulting in limited calculation accuracy of the electron impact ionization cross-section. Summary of the Invention

[0009] Based on this, the technical solution provided by the present invention combines the easily calculable atomic structure code (Flexible Atomic Code, FAC) program with the Hypernetted-Chain (HNC) approximation, comprehensively considers the free electron screening effect and the ion-ion coupling effect in the plasma, improves the calculation accuracy, and solves the deviation problem existing in the traditional calculation method of electron impact ionization cross-sections in hot and dense plasmas.

[0010] To achieve the above object, the present invention provides a method for calculating the electron impact ionization cross section of hot dense plasma, comprising the following steps: S1: Using the HNC approximation to calculate the correlation functions of the free electron-free electron, electron-ion, and ion-ion coupling models; S2: Introducing the correlation functions into the FAC program to calculate the bound electron wave functions of the target element in the plasma environment; S3: Based on the bound electron wave functions in the plasma environment, using the distorted wave approximation to calculate the electron impact ionization cross section of the target element in the plasma environment; the correlation functions include the pair correlation function and the direct correlation function.

[0011] Specifically, step S1 includes:

[0012] S11: Under the HNC approximation condition, establish the following correlation function relationship according to the free electron-free electron, electron-ion, and ion-ion coupling models: ;

[0013] Wherein, is the pair correlation function of a particles and b particles, is the reciprocal of the combined temperature function of a particles and b particles, is the interaction potential between a particles and b particles, is the direct correlation function of a particles and b particles, is the distance between a particles and b particles.

[0014] S12: Self-consistently iterate to solve the correlation functions according to the Ornstein-Zernike (OZ) relation.

[0015] Furthermore, step S2 specifically includes:

[0016] S21: In the FAC program of the atomic structure, add the plasma screening potential to the central field single-electron effective potential to construct the central field potential energy in the plasma environment.

[0017] S22: Solve the central field potential energy and the bound electron wave functions in the plasma environment by the self-consistent iteration method.

[0018] Wherein, the central field potential energy in the plasma environment is: ;

[0019] is the central field single-electron effective potential, is the screening potential between electrons and ions, is the screening potential of the interaction between ions on electrons.

[0020] Further, step S22 specifically includes self - consistently solving the bound electron wave function according to the single - electron Dirac equation in a plasma environment, where the bound electron wave function includes a large component and a small component.

[0021] Preferably, the distorted - wave approximation is adopted to calculate the electron - impact ionization cross - section in a plasma environment.

[0022] Preferably, the target elements include carbon, nitrogen, and oxygen elements.

[0023] The beneficial effects achieved by the present invention through the above - mentioned technical solutions are as follows:

[0024] 1) By using the correlation functions between electrons - electrons and electrons - ions in the HNC approximation, the free - electron screening effect and the ion - ion coupling effect in the plasma environment are implanted into the FAC central - potential - field model, reducing the calculation error of the free - atom model and improving the calculation accuracy.

[0025] 2) The technical solution provided by the present invention, through the calculation of the electron - impact ionization cross - sections of carbon, nitrogen, and oxygen atoms at the solar radiation / convection boundary in the embodiment, reveals the ionization process of high - temperature and dense plasmas. This not only helps to improve the theoretical model of solar radiation opacity and enhance the understanding of the energy - transport process inside stars, but also can be extended to the model calculations of astrophysical environments such as inertial confinement fusion or white dwarfs. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 is a schematic flow chart of the calculation method of the electron - impact ionization cross - section according to an embodiment of the present invention.

[0027] Figure 2 is the spatial distribution of the bound - electron radial wave function according to an embodiment of the present invention.

[0028] Figure 3 is a comparison diagram of the calculation results of the electron - impact ionization cross - section varying with the incident - electron energy according to an embodiment of the present invention.

[0029] Figure 4 is a diagram of the calculation results of the electron - impact ionization cross - sections of different valence states of different elements according to an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0030] In order to make the objectives, technical solutions, and advantages of the present invention clearer and more understandable, the specific implementation methods of the present invention will be further described in detail below with reference to the embodiments. It should be understood that the embodiments described herein are only used to explain the present invention, but not to limit the scope of the present invention.

[0031] Please refer to the atta Figure 1 , Figure 1It is a schematic flowchart of the calculation method for the electron impact ionization cross-section of an embodiment of the present invention. As can be seen from the figure, the steps of this calculation method include: S1: Using the HNC approximation to calculate the correlation functions of the free electron-free electron, electron-ion, and ion-ion coupling models; S2: Introducing the correlation function into the FAC program to calculate the bound electron wave function of the target element in the plasma environment; S3: Based on the bound electron wave function in the plasma environment, using the distorted wave approximation to calculate the electron impact ionization cross-section of the target element in the plasma environment; the correlation function includes the pair correlation function and the direct correlation function.

[0032] In an embodiment of the present invention, the target elements include carbon, nitrogen, and oxygen elements.

[0033] Step S1 specifically includes:

[0034] S11: Under the HNC approximation condition, establish the following correlation function relationship according to the free electron-free electron, electron-ion, and ion-ion coupling models: ;

[0035] Wherein, is the pair correlation function of a particles and b particles, is the reciprocal of the combined temperature function of a particles and b particles, is the interaction potential between a particles and b particles, is the direct correlation function of a particles and b particles, is the distance between a particles and b particles.

[0036] S12: Self-consistently iterate to solve the correlation function according to the Ornstein-Zernike (OZ) relation. In particular, the OZ relation is specifically: ;

[0037] Wherein, is the density of c particles.

[0038] Step S2 specifically includes:

[0039] S21: In the FAC program of the atomic structure, add the plasma screening potential to the central field single-electron effective potential to construct the central field potential energy in the plasma environment. In the embodiment of the present invention, the central field potential energy in the plasma environment is expressed as: ;

[0040] is the central field single-electron effective potential, is the screening potential between electrons and ions, Brings a shielding potential for electrons to the interaction between ions.

[0041] Specifically, the shielding potential between electrons and ions and the shielding potential between ions are expressed as: ;

[0042] Wherein, is the free electron density, is the ion density.

[0043] S22: Solve the central field single-electron effective potential and the bound electron wave function in the plasma environment by the self-consistent iteration method.

[0044] Furthermore, step S22 specifically includes self-consistently solving the bound electron wave function according to the single-electron Dirac equation in the plasma environment, wherein the bound electron wave function includes a large component and a small component.

[0045] Specifically, the single-electron Dirac equation in the plasma environment is expressed as: ;

[0046] Wherein, is the fine structure constant, is the large component of the radial wave function, is the small component of the radial wave function, is the orbital energy eigenvalue.

[0047] In one or other embodiments of the present invention, the bound electron density is also calculated: ;

[0048] Wherein, is the bound electron density, is the occupation number of the state. For a uniformly distributed free electron density through the non-uniformly distributed free electron density at the boundary of the ion sphere is given. The non-uniformly distributed free electron density is obtained under the Thomas-Fermi approximation while considering the Fermi-Dirac distribution: ;

[0049] Wherein, , c and are the speed of light and the chemical potential respectively. For an ion with a nuclear charge of Z, the chemical potential is determined by the electrical neutrality of the ion sphere: 。

[0050] In the embodiments of the present invention, the distorted wave approximation is adopted to calculate the electron impact ionization cross section in a plasma environment. In particular, the specific form of the electron impact ionization cross section obtained in a plasma environment is: ;

[0051] wherein, is the momentum of the incident electron, is the energy of the incident electron, is the statistical weight of the initial state of the target ion, is the relativistic angular quantum number of the incident electron, is the relativistic angular quantum number of the scattered electron, is the relativistic angular quantum number of the outgoing electron, is the total angular momentum of the system when the continuum electron is coupled with the target ion, is the projection of the total angular momentum of the system when the continuum electron is coupled with the target ion in the z direction, is the initial state wave function of the target ion, is the final state wave function of the target ion.

[0052] Please refer to Appendix Figure 2 , Figure 2 is the spatial distribution of the bound electron radial wave function in the embodiments of the present invention. In the embodiments of the present invention, what is investigated is the of configuration of bound electron radial wave function. In the case of an isolated atom, the wave function is obtained through the relativistic Dirac equation: ;

[0053] wherein, the relativistic Hamiltonian of an atom or ion with N electrons is: .

[0054] And the basis state is obtained by the antisymmetric sum of the product of N single-electron Dirac spinors : ;

[0055] Here, is the principal quantum number of the electron, is the relativistic angular quantum number, is the magnetic quantum number, and It is a two-component spherical harmonic spinor. The large component P and the small component Q of the radial wave function are obtained by self-consistently iteratively solving the single-electron Dirac equation in a plasma environment.

[0056] Among them, Figure 2 (a) shows the radial wave functions in the case of isolated ions and when the temperature T is 100 eV and the electron number densities are respectively , , cases. Figure 2 (b) shows the radial wave functions in the case of isolated ions and when the electron number density is , and the temperature T is 50 eV, 100 eV, and 180 eV respectively. In the embodiments of the present invention, the screening potential of ions and free electrons in the plasma environment is calculated by the HNC approximation. As can be seen from the figure, compared with the bound electron wave function in the isolated case, the bound electron wave function in the plasma environment extends farther outwards, that is, the attraction of the ion core to the bound electron in the plasma environment becomes weaker. At a certain plasma temperature, the higher the plasma density, the more obvious the outward extension of the bound electron wave function; at a certain plasma density, the lower the plasma temperature, the more obvious the outward extension of the bound electron wave function. This is because the lower the plasma temperature and the higher the density, the stronger the coupling of charged particles in the plasma environment, the stronger the screening brought by the environmental effect, and thus the greater the change in the electronic structure.

[0057] Please refer to Appendix Figure 3 , Figure 3 which is a comparison diagram of the calculated results of the electron impact ionization cross-section varying with the incident electron energy in the embodiments of the present invention. In the embodiments of the present invention, what is examined is the of electron impact ionization cross-section varying with the incident electron energy. Among them, Figure 3 (a) shows the calculated results of the electron impact ionization cross-section in the case of isolated ions and when the temperature T is 100 eV and the electron number densities are respectively , , cases, Figure 3 (b) shows the calculated results of the electron impact ionization cross-section in the case of isolated ions and when the electron number density is , Calculation results of electron impact ionization cross sections at temperatures T of 50 eV, 100 eV, and 180 eV respectively. As can be seen from the figure, the screening effect in the plasma environment not only causes the threshold energy of the cross section to decrease, but also leads to an enhancement of the cross section. And at a certain plasma temperature, as the plasma density increases, the more the threshold energy decreases, and the greater the enhancement of the electron impact ionization cross section. At a given plasma density, as the plasma temperature decreases, the more the threshold energy decreases, and the greater the enhancement of the electron impact ionization cross section. Because with the enhancement of the plasma coupling strength, the attraction of the ion core to the bound electrons becomes weaker and weaker, resulting in the easier occurrence of the electron impact ionization process. From Figure 3 (b), it can be found that in the plasma environment at the boundary of the solar radiation / convection region, the obtained electron impact ionization cross section is approximately twice that in the case of isolated ions, indicating that the plasma environment effect in this solar region has a significant impact on electron impact ionization.

[0058] Please refer to the appendix Figure 4 , Figure 4 is the graph of the calculation results of electron impact ionization cross sections for different valence states of various elements in the embodiment of the present invention. In the embodiment of the present invention, what is investigated is , at the solar radiation convection region, the electron impact ionization cross section with the change of incident electron energy. Among them, Figure 4 (a) is the , , calculation results, Figure 4 (b) is the , , , calculation results, Figure 4 (c) is the , , , , calculation results, Figure 4 (d) is the , , calculation results. Since the electron impact ionization cross sections of these elements in high valence states are very small, in the embodiment of the present invention, only For ions of shell electrons, the corresponding collision ionization cross-sections are not given. As can be seen from the figure, for the same element, as the ionic valence state gradually increases, the threshold energy of the electron collision ionization cross-section becomes larger and the cross-section becomes smaller. In the same plasma environment, as the ionic valence state increases, the shielding effect of the bound electrons on the nucleus becomes smaller, resulting in a stronger attraction of the nucleus to the bound electrons. Therefore, the electron collision ionization process becomes more and more difficult to occur. For ions with the same number of bound electrons but different elements, as the nuclear charge number gradually increases, the threshold energy of electron collision ionization gradually increases and the cross-section gradually decreases. The increase in the nuclear charge number will lead to an enhanced attraction of the nucleus to the bound electrons, and the electron collision ionization process will become more and more difficult to occur.

[0059] As can be seen from Figure 4 (a), the electron collision ionization cross-section of is about 2.5 times that of and 12.5 times that of Figure 4 As can be seen from (b), the electron collision ionization cross-section of is about 5 times that of Figure 4 As can be seen from (c), the electron collision ionization cross-section of is about 3 times that of 11 times that of and 60 times that of

[0060] Finally, it should be noted that the above specific implementation methods are only used to illustrate the technical solutions of the present invention and are not restrictive. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the technical solutions of the present invention, and they should all be covered by the scope of the claims of the present invention.

Claims

1. A method for calculating the electron impact ionization cross section of a hot dense plasma, characterized in that: The following steps are involved: S1: The correlation functions of the free electron-free electron, electron-ion and ion-ion coupling models are calculated using the HNC approximation; S2: introducing the correlation function into the FAC program to calculate the bound electron wave function of the target element in the plasma environment; S3: Based on the bound electron wave function in the plasma environment, the electron impact ionization cross section of the target element in the plasma environment is calculated using the distorted wave approximation; The correlation function includes a pair correlation function and a direct correlation function.

2. The method for calculating the electron impact ionization cross section according to claim 1, characterized in that: Step S1 specifically includes: S11: Under the HNC approximation, the correlation function relationship is established according to the free electron-free electron, electron-ion and ion-ion coupling models as follows: ， in, is the pair correlation function of particle a and particle b, is the inverse of the combined temperature function of particles a and b, is the interaction potential between particles a and b, is the direct correlation function between particle a and particle b, is the distance between particle a and particle b; S12: Solve the correlation function according to the OZ relationship through self-consistent iteration.

3. The method for calculating the electron impact ionization cross section according to claim 2, characterized in that: Step S2 specifically includes: S21: In the FAC program of atomic structure, plasma shielding potential is added to the central field single electron effective potential to construct the central field potential energy in plasma environment; S22: solving the central field potential energy and bound electron wave function in the plasma environment by a self-consistent iteration method; Wherein, the central field potential energy in the plasma environment is: ， is the central field single electron effective potential, is the screening potential between electrons and ions, It brings screening potential for electrons in ion-ion interactions.

4. The method for calculating the electron impact ionization cross section according to claim 3, characterized in that: Step S22 specifically includes self-consistently solving the bound electron wave function according to the single electron Dirac equation in a plasma environment, wherein the bound electron wave function includes a large component and a small component.

5. The method for calculating the electron impact ionization cross section according to claim 1, characterized in that: The target elements include carbon, nitrogen and oxygen.

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